The main purpose of this paper is to study north–south Smale solenoid diffeomorphisms on $3$-manifolds by using affine Hirsch foliations. A north–south Smale solenoid diffeomorphism $f$ on a closed $3$-manifold $M$ is a diffeomorphism whose non-wandering set is composed of a Smale solenoid attractor $\unicode[STIX]{x1D6EC}_{a}$ and a Smale solenoid repeller $\unicode[STIX]{x1D6EC}_{r}$. The key observation is that a north–south Smale solenoid diffeomorphism $f$ automatically induces two non-isotopically leaf-conjugate affine Hirsch foliations ${\mathcal{H}}^{s}$ and ${\mathcal{H}}^{u}$ on the orbit space of the wandering set of $f$ (abbreviated to the wandering orbit space of $f$) by the stable and unstable manifolds of $\unicode[STIX]{x1D6EC}_{a}$ and $\unicode[STIX]{x1D6EC}_{r}$, respectively. Under this viewpoint, we build some close relationships between north–south Smale solenoid diffeomorphisms and Hirsch manifolds (the closed $3$-manifolds admitting two non-isotopically leaf-conjugate affine Hirsch foliations).

• ∙ On the one hand, the union of the wandering orbit spaces is nearly in one-to-one correspondence with the union of Hirsch manifolds.

• ∙ On the other hand, surprisingly, the topology of the wandering orbit space (Hirsch manifold) is nearly a complete invariant of north–south Smale solenoid diffeomorphisms up to semi-global conjugacy.

Moreover, as applications, we consider several more concrete questions. For instance, we prove that every diffeomorphism in many semi-global conjugacy classes of north–south Smale solenoid diffeomorphisms are not structurally stable.

Let $M$ be an analytic connected 2-manifold with empty boundary, over the ground field $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$. Let $Y$ and $X$ denote differentiable vector fields on $M$. We say that $Y$ tracks $X$ if $[Y,X]=fX$ for some continuous function $f:\,M\rightarrow \mathbb{F}$. A subset $K$ of the zero set $\mathsf{Z}(X)$ is an essential block for $X$ if it is non-empty, compact and open in $\mathsf{Z}(X)$, and the Poincaré–Hopf index $\mathsf{i}_{K}(X)$ is non-zero. Let ${\mathcal{G}}$ be a finite-dimensional Lie algebra of analytic vector fields that tracks a non-trivial analytic vector field $X$. Let $K\subset \mathsf{Z}(X)$ be an essential block. Assume that if $M$ is complex and $\mathsf{i}_{K}(X)$ is a positive even integer, no quotient of ${\mathcal{G}}$ is isomorphic to $\mathfrak{s}\mathfrak{l}(2,\mathbb{C})$. Then ${\mathcal{G}}$ has a zero in $K$ (main result). As a consequence, if $X$ and $Y$ are analytic, $X$ is non-trivial, and $Y$ tracks $X$, then every essential component of $\mathsf{Z}(X)$ meets $\mathsf{Z}(Y)$. Fixed-point theorems for certain types of transformation groups are proved. Several illustrative examples are given.

We show that a germ of a holomorphic one-dimensional foliation at a singularity in a space of dimension two admits a holomorphic first integral if and only if there are infinitely many closed leaves and a finite number of separatrices, with each separatrix having linearizable holonomy. Indeed, if there are infinitely many closed leaves and the set of separatrices is finite, then the foliation admits either a holomorphic first integral or a formal simple integrating factor of Darboux type.

We study the center problem for the class ${\mathcal{E}}_{{\rm\Gamma}}$ of Abel differential equations $dv/dt=a_{1}v^{2}+a_{2}v^{3}$, $a_{1},a_{2}\in L^{\infty }([0,T])$, such that images of Lipschitz paths $\tilde{A}:=(\int _{0}^{\cdot }a_{1}(s)\,ds,\int _{0}^{\cdot }a_{2}(s)\,ds):[0,T]\rightarrow \mathbb{R}^{2}$ belong to a fixed compact rectifiable curve ${\rm\Gamma}$. Such a curve is said to be universal if whenever an equation in ${\mathcal{E}}_{{\rm\Gamma}}$ has center on $[0,T]$, this center must be universal, i.e. all iterated integrals in coefficients $a_{1},a_{2}$ of this equation must vanish. We investigate some basic properties of universal curves. Our main results include an algebraic description of a universal curve in terms of a certain homomorphism of its fundamental group into the group of locally convergent invertible power series with the product being the composition of series, explicit examples of universal curves, and approximation of Lipschitz triangulable curves by universal ones.

Given a factor code between sofic shifts X and Y, there is a family of decompositions of the original code into factor codes such that the entropies of the intermediate subshifts arising from the decompositions are dense in the interval from the entropy of Y to that of X. Furthermore, if X is of finite type, we can choose those intermediate subshifts as shifts of finite type. In the second part of the paper, given an embedding from a shift space to an irreducible sofic shift, we characterize the set of the entropies of the intermediate subshifts arising from the decompositions of the given embedding into embeddings.

We consider piecewise $C^{2}$ non-flat maps of the interval and show that, for Lebesgue almost every point, its omega-limit set is either a periodic orbit, a cycle of intervals or the closure of the orbits of a subset of the critical points. In particular, every piecewise $C^{2}$ non-flat map of the interval displays only a finite number of non-periodic attractors.

In this paper we consider symplectic and volume-preserving diffeomorphisms of compact manifolds, and prove that, C^{r} generically, 1\leq r\leq\infty, the invariant manifolds of hyperbolic periodic points contain a dense subset of recurrent points. As a consequence, each of these invariant manifolds is contained in its omega limit set, and its closure is a chain transitive set.

We consider continuous cocycles arising from CMV and Jacobi matrices. Assuming that the Verblunsky and Jacobi coefficients arise from generalized skew-shifts, we prove that uniform hyperbolicity of the associated cocycles is $C^0$ -dense. This implies that the associated CMV and Jacobi matrices have a Cantor spectrum for a generic continuous sampling map.

A smooth $\mathbb{Z}^p$-action on a closed orientable manifold M is cohomologically rigid (CR) if the cohomology of the induced $\mathbb{Z}^p$-module on the smooth functions on M is isomorphic to the real cohomology of the torus Tp. We prove some general properties of a smooth $\mathbb{Z}^p$-action whose first cohomology is isomorphic to Rp. When the manifold is a low-dimensional torus Tq, $1 \le q \le 2$, we prove that any minimal (all orbits are dense) smooth $\mathbb{Z}^p$-action on Tq whose first cohomology is isomorphic to Rp is $C^\infty$ conjugate to an affine $\mathbb{Z}^p$-action. As a corollary we show that the CR $\mathbb{Z}^p$-actions on Tq, $1 \le q \le 2$, are smooth conjugations of affine CR actions.

We are interested in attracting sets of $\mathbb{P}^{k}(\mathbb{C})$ which are of small topological degree and of codimension one. We first show that there exists a large family of examples. Then we study their ergodic and pluripotential theoretic properties.

We consider ℝn skew-products of a class of hyperbolic dynamical systems. It was proved by Niţică and Pollicott [Transitivity of Euclidean extensions of Anosov diffeomorphisms. Ergod. Th. & Dynam. Sys. 25 (2005), 257–269] that for an Anosov diffeomorphism ϕ of an infranilmanifold Λ there is (subject to avoiding natural obstructions) an open and dense set f:Λ→ℝN for which the skew-product ϕf(x,v)=(ϕ(x),v+f(x)) on Λ×ℝN has a dense orbit. We prove a similar result in the context of an Axiom A hyperbolic flow on an attractor.

A classical approach to the study of a dynamical system is to code it using a partition. This leads to the question of the quality of this coding and in particular to the problem of finding checkable conditions ensuring that the coding is essentially one-to-one. We prove that, with respect to an invariant measure with only positive Lyapunov exponents, it is basically enough that the map be one-to-one on each piece of the partition. This is a multi-dimensional generalization of a well-known fact in one-dimensional dynamics. We apply this result to the study of the measure of maximal entropy of endomorphisms of complex projective spaces.

In this paper we consider a quasi-linear approximation for the 3D Euler system as well as for the Navier–Stokes system in our previous paper Moscow Math. J.1(3) (2001), 381–388). We define finite-dimensional versions of the approximation and study properties of their solutions.

If K is a countable amenable group acting freely and ergodically on a probability space (Γ, μ), and G is an arbitrary countable amenable group, we construct an injection of the space of unitary representations of G into the space of unitary 1-cocyles for K on (Γ, μ); this injection preserves intertwining operators. We apply this to show that for many of the standard non-type-I amenable groups H, the representation theory of H contains that of every countable amenable group.

We discuss a metric description of the divergence of a (projectively) Anosov flow in dimension 3, in terms of its associated expansion rates and give metric and contact geometric characterizations of when a projectively Anosov flow is Anosov. We then study the symmetries that the existence of an invariant volume form yields on the geometry of an Anosov flow, from various viewpoints of the theory of contact hyperbolas, Reeb dynamics, and Liouville geometry, and give characterizations of when an Anosov flow is volume preserving, in terms of those theories. We finally use our study to show that the bi-contact surgery operations of Salmoiraghi [Surgery on Anosov flows using bi-contact geometry. Preprint, 2021, arXiv:2104.07109; and Goodman surgery and projectively Anosov flows. Preprint, 2022, arXiv:2202.01328] can be applied in an arbitrary small neighborhood of a periodic orbit of any Anosov flow. In particular, we conclude that the Goodman surgery of Anosov flows can be performed using the bi-contact surgery operations of Salmoiraghi [Goodman surgery and projectively Anosov flows. Preprint, 2022, arXiv:2202.01328].

Let G be the group $\text {PAff}_{+}(\mathbb R/\mathbb Z)$ of piecewise affine circle homeomorphisms or the group ${\operatorname {\mathrm {Diff}}}^{{\kern1pt}\infty }(\mathbb R/\mathbb Z)$ of smooth circle diffeomorphisms. A constructive proof that all irrational rotations are distorted in G is given.

In this paper, we reduce the logarithmic Sarnak conjecture to the $\{0,1\}$-symbolic systems with polynomial mean complexity. By showing that the logarithmic Sarnak conjecture holds for any topologically dynamical system with sublinear complexity, we provide a variant of the $1$-Fourier uniformity conjecture, where the frequencies are restricted to any subset of $[0,1]$ with packing dimension less than one.

In this paper it is shown that for a minimal system $(X,T)$ and $d,k\in \mathbb {N}$ , if $(x,x_{i})$ is regionally proximal of order d for $1\leq i\leq k$ , then $(x,x_{1},\ldots ,x_{k})$ is $(k+1)$ -regionally proximal of order d. Meanwhile, we introduce the notion of $\mathrm {IN}^{[d]}$ -pair: for a dynamical system $(X,T)$ and $d\in \mathbb {N}$ , a pair $(x_{0},x_{1})\in X\times X$ is called an $\mathrm {IN}^{[d]}$ -pair if for any $k\in \mathbb {N}$ and any neighborhoods $U_{0} ,U_{1} $ of $x_{0}$ and $x_{1}$ respectively, there exist different $(p_{1}^{(i)},\ldots ,p_{d}^{(i)})\in \mathbb {N}^{d} , 1\leq i\leq k$ , such that

$$ \begin{align*} \bigcup_{i=1}^{k}\{ p_{1}^{(i)}\epsilon(1)+\cdots+p_{d}^{(i)} \epsilon(d):\epsilon(j)\in \{0,1\},1\leq j\leq d\}\backslash \{0\}\in \mathrm{Ind}(U_{0},U_{1}), \end{align*} $$

$\mathrm {Ind}(U_{0},U_{1})$

$(U_{0},U_{1})$

$\mathrm {IN}^{[d]}$

Let f:Tn→Tn (n ≥ 3) be a hyperbolic toral automorphism. Let A be the set of α > 0 such that there is a Hölder continuous path of index α in Tn with 1-dimensional orbit-closure under f We prove that α0 = sup A can be expressed in terms of the eigenvalues of f and that α0 ∈ A if and only if α0 < 1.